Is there a branch of Mathematics which connects Calculus and Discrete Math / Number Theory?

The branch of math which connects calculus and combinatorics is called the theory of generating functions, or more evocatively analytic combinatorics. This theory is often used to find asymptotics of combinatorial sequences, such as might occur in the analysis of the running time of some algorithm. Here is a free textbook about it.

The branch of math which connects calculus and number theory is called analytic number theory.

This is certainly possible. Here is a concrete example that seems to perfectly fit the description "mapping this discrete problem onto a continuous one and using the power of calculus to solve it, finally extracting out the answer of the original discrete problem."

The Ramsey number $r(3, n)$ is defined to be the least natural $N$ with the property that among any $N$ people there are always either three, all of whom know each other, or $n$, none of whom know each other. This is a purely discrete consideration.

Now the asymptotic order of $r(3, n)$ is $\frac{n^2}{\log n}$ and the upper bound proof follows easily from a graph-theoretic fact that is essentially proven by studying the analytic properties of a certain real function.

However, the proof is a bit of an accumulation of little lucky accidents, but is perfectly accessible if you know basic analysis and basic graph theory. You can find a nice exposition here https://www.dpmms.cam.ac.uk/~dc340/RamseyLecture3.pdf

Calculus can occasionally be used in extremal combinatorics in order to find a maximizing or minimizing solution to a problem. The only difference is that one must restrain ones solutions in order to get a discrete answer, as opposed to taking any real solution you might find in calculus.